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Scale recurrence lemma and dimension formula for Cantor sets in the complex plane
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 44 / Issue 11 / November 2024
- Published online by Cambridge University Press:
- 25 March 2024, pp. 3316-3357
- Print publication:
- November 2024
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We prove a multidimensional conformal version of the scale recurrence lemma of Moreira and Yoccoz [Stable intersections of regular Cantor sets with large Hausdorff dimensions. Ann. of Math. (2) 154(1) (2001), 45–96] for Cantor sets in the complex plane. We then use this new recurrence lemma, together with Moreira’s ideas in [Geometric properties of images of Cartesian products of regular Cantor sets by differentiable real maps. Math. Z. 303 (2023), 3], to prove that under the right hypothesis for the Cantor sets
$K_1,\ldots ,K_n$ and the function 
$h:\mathbb {C}^{n}\to \mathbb {R}^{l}$, the following formula holds: 
$$ \begin{align*}HD(h(K_1\times K_2 \times \cdots\times K_n))=\min \{l,HD(K_1)+\cdots+HD(K_n)\}.\end{align*} $$
